A traveller on the planet of liars and truth tellers met four people and asked them: “Who are you?”. They received the following answers:
1st: “We are all liars.”
2nd: “Among us is exactly one liar.”
3rd: “Among us there are two liars.”
4th: “I have never lied and I’m not lying”.
The traveller quickly realised who the fourth resident was. How did they do it?
In the lower left corner of an 8 by 8 chessboard is a chip. Two in turn move it one cell up, right or right-up diagonally. The one who puts the chip in the upper right corner wins. Who will win in a regular game?
a) There are 10 coins. It is known that one of them is fake (by weight, it is heavier than the real ones). How can you determine the counterfeit coin with three weighings on scales without weights?
b) How can you determine the counterfeit coin with three weighings, if there are 27 coins?
Find a natural number greater than one that occurs in the Pascal triangle a) more than three times; b) more than four times.
Prove there are no integer solutions for the equation \(3x^2 + 2 = y^2\).
Prove that the sum of
a) any number of even numbers is even;
b) an even number of odd numbers is even;
c) an odd number of odd numbers is odd.
Prove that the product of
a) two odd numbers is odd;
b) an even number with any integer is even.
On the selection to the government of the planet of liars and truth tellers \(12\) candidates gave a speech about themselves. After a while, one said: “before me only once did someone lie” Another said: “And now-twice.” “And now – thrice” – said the third, and so on until the \(12\)th, who said: “And now \(12\) times someone has lied.” Then the presenter interrupted the discussion. It turned out that at least one candidate correctly counted how many times someone had lied before him. So how many times have the candidates lied?
Two people play the following game. Each player in turn rubs out 9 numbers (at his choice) from the sequence \(1, 2, \dots , 100, 101\). After eleven such deletions, 2 numbers will remain. The first player is awarded so many points, as is the difference between these remaining numbers. Prove that the first player can always score at least 55 points, no matter how played the second.
A six-digit phone number is given. How many seven-digit numbers are there from which one can obtain this six-digit number by deleting one digit?